Optimal throwing angle

Percentage Accurate: 67.1% → 99.6%

Time: 4.2s

Alternatives: 7

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))

C

double code(double v, double H) {
	return atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, h)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((v / sqrt(((v * v) - ((2.0d0 * 9.8d0) * h)))))
end function

Java

public static double code(double v, double H) {
	return Math.atan((v / Math.sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Python

def code(v, H):
	return math.atan((v / math.sqrt(((v * v) - ((2.0 * 9.8) * H)))))

Julia

function code(v, H)
	return atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
end

MATLAB

function tmp = code(v, H)
	tmp = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
end

Wolfram

code[v_, H_] := N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))

C

double code(double v, double H) {
	return atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, h)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((v / sqrt(((v * v) - ((2.0d0 * 9.8d0) * h)))))
end function

Java

public static double code(double v, double H) {
	return Math.atan((v / Math.sqrt(((v * v) - ((2.0 * 9.8) * H)))));
}

Python

def code(v, H):
	return math.atan((v / math.sqrt(((v * v) - ((2.0 * 9.8) * H)))))

Julia

function code(v, H)
	return atan(Float64(v / sqrt(Float64(Float64(v * v) - Float64(Float64(2.0 * 9.8) * H)))))
end

MATLAB

function tmp = code(v, H)
	tmp = atan((v / sqrt(((v * v) - ((2.0 * 9.8) * H)))));
end

Wolfram

code[v_, H_] := N[ArcTan[N[(v / N[Sqrt[N[(N[(v * v), $MachinePrecision] - N[(N[(2.0 * 9.8), $MachinePrecision] * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{elif}\;v \leq 4 \cdot 10^{+107}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -4e+154)
   (atan -1.0)
   (if (<= v 4e+107) (atan (/ v (sqrt (fma v v (* -19.6 H))))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -4e+154) {
		tmp = atan(-1.0);
	} else if (v <= 4e+107) {
		tmp = atan((v / sqrt(fma(v, v, (-19.6 * H)))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -4e+154)
		tmp = atan(-1.0);
	elseif (v <= 4e+107)
		tmp = atan(Float64(v / sqrt(fma(v, v, Float64(-19.6 * H)))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -4e+154], N[ArcTan[-1.0], $MachinePrecision], If[LessEqual[v, 4e+107], N[ArcTan[N[(v / N[Sqrt[N[(v * v + N[(-19.6 * H), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -4.00000000000000015e154

   1. Initial program 3.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \tan^{-1} \color{blue}{-1} \]

   if -4.00000000000000015e154 < v < 3.9999999999999999e107

   1. Initial program 99.8%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v - \left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \color{blue}{\left(2 \cdot \frac{49}{5}\right) \cdot H}}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v + \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H}}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{v \cdot v} + \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H}}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, \left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H\right)}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{49}{5}\right)\right) \cdot H}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{49}{5}}\right)\right) \cdot H\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \left(\mathsf{neg}\left(\color{blue}{\frac{98}{5}}\right)\right) \cdot H\right)}}\right) \]

      9. metadata-eval 99.8

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\mathsf{fma}\left(v, v, \color{blue}{-19.6} \cdot H\right)}}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\mathsf{fma}\left(v, v, -19.6 \cdot H\right)}}}\right) \]

   if 3.9999999999999999e107 < v

   1. Initial program 22.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)\\ \mathbf{elif}\;v \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{-19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.6e-58)
   (atan (fma -9.8 (/ H (* v v)) -1.0))
   (if (<= v 5.2e-103)
     (atan (/ v (sqrt (* -19.6 H))))
     (atan (/ v (fma (/ H v) -9.8 v))))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.6e-58) {
		tmp = atan(fma(-9.8, (H / (v * v)), -1.0));
	} else if (v <= 5.2e-103) {
		tmp = atan((v / sqrt((-19.6 * H))));
	} else {
		tmp = atan((v / fma((H / v), -9.8, v)));
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6.6e-58)
		tmp = atan(fma(-9.8, Float64(H / Float64(v * v)), -1.0));
	elseif (v <= 5.2e-103)
		tmp = atan(Float64(v / sqrt(Float64(-19.6 * H))));
	else
		tmp = atan(Float64(v / fma(Float64(H / v), -9.8, v)));
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.6e-58], N[ArcTan[N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 5.2e-103], N[ArcTan[N[(v / N[Sqrt[N[(-19.6 * H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(v / N[(N[(H / v), $MachinePrecision] * -9.8 + v), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.60000000000000052e-58

   1. Initial program 54.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - \color{blue}{1 \cdot 1}\right) \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)} \]

      3. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1} \cdot 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{{v}^{2}}, -1\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \color{blue}{\frac{H}{{v}^{2}}}, -1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

      8. lower-*.f64 92.9

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)} \]

   if -6.60000000000000052e-58 < v < 5.19999999999999993e-103

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\frac{-98}{5} \cdot H}}}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 93.3

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{-19.6 \cdot H}}}\right) \]

   5. Applied rewrites 93.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{-19.6 \cdot H}}}\right) \]

   if 5.19999999999999993e-103 < v

   1. Initial program 56.0%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in H around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{v + \frac{-49}{5} \cdot \frac{H}{v}}}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{-49}{5} \cdot \frac{H}{v} + v}}\right) \]

      2. *-commutative N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\frac{H}{v} \cdot \frac{-49}{5}} + v}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\mathsf{fma}\left(\frac{H}{v}, \frac{-49}{5}, v\right)}}\right) \]

      4. lower-/.f64 85.9

         \[\leadsto \tan^{-1} \left(\frac{v}{\mathsf{fma}\left(\color{blue}{\frac{H}{v}}, -9.8, v\right)}\right) \]

   5. Applied rewrites 85.9%

      \[\leadsto \tan^{-1} \left(\frac{v}{\color{blue}{\mathsf{fma}\left(\frac{H}{v}, -9.8, v\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)\\ \mathbf{elif}\;v \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\tan^{-1} \left(\frac{v}{\sqrt{-19.6 \cdot H}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.6e-58)
   (atan (fma -9.8 (/ H (* v v)) -1.0))
   (if (<= v 5.2e-103) (atan (/ v (sqrt (* -19.6 H)))) (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.6e-58) {
		tmp = atan(fma(-9.8, (H / (v * v)), -1.0));
	} else if (v <= 5.2e-103) {
		tmp = atan((v / sqrt((-19.6 * H))));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6.6e-58)
		tmp = atan(fma(-9.8, Float64(H / Float64(v * v)), -1.0));
	elseif (v <= 5.2e-103)
		tmp = atan(Float64(v / sqrt(Float64(-19.6 * H))));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.6e-58], N[ArcTan[N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 5.2e-103], N[ArcTan[N[(v / N[Sqrt[N[(-19.6 * H), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.60000000000000052e-58

   1. Initial program 54.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - \color{blue}{1 \cdot 1}\right) \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)} \]

      3. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1} \cdot 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{{v}^{2}}, -1\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \color{blue}{\frac{H}{{v}^{2}}}, -1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

      8. lower-*.f64 92.9

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)} \]

   if -6.60000000000000052e-58 < v < 5.19999999999999993e-103

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{\frac{-98}{5} \cdot H}}}\right) \]
   4. Step-by-step derivation

      1. lower-*.f64 93.3

         \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{-19.6 \cdot H}}}\right) \]

   5. Applied rewrites 93.3%

      \[\leadsto \tan^{-1} \left(\frac{v}{\sqrt{\color{blue}{-19.6 \cdot H}}}\right) \]

   if 5.19999999999999993e-103 < v

   1. Initial program 56.0%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.4%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)\\ \mathbf{elif}\;v \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -6.6e-58)
   (atan (fma -9.8 (/ H (* v v)) -1.0))
   (if (<= v 5.2e-103)
     (atan (* (sqrt (/ -0.05102040816326531 H)) v))
     (atan 1.0))))

C

double code(double v, double H) {
	double tmp;
	if (v <= -6.6e-58) {
		tmp = atan(fma(-9.8, (H / (v * v)), -1.0));
	} else if (v <= 5.2e-103) {
		tmp = atan((sqrt((-0.05102040816326531 / H)) * v));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -6.6e-58)
		tmp = atan(fma(-9.8, Float64(H / Float64(v * v)), -1.0));
	elseif (v <= 5.2e-103)
		tmp = atan(Float64(sqrt(Float64(-0.05102040816326531 / H)) * v));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -6.6e-58], N[ArcTan[N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[v, 5.2e-103], N[ArcTan[N[(N[Sqrt[N[(-0.05102040816326531 / H), $MachinePrecision]], $MachinePrecision] * v), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.60000000000000052e-58

   1. Initial program 54.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - \color{blue}{1 \cdot 1}\right) \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)} \]

      3. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1} \cdot 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{{v}^{2}}, -1\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \color{blue}{\frac{H}{{v}^{2}}}, -1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

      8. lower-*.f64 92.9

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)} \]

   if -6.60000000000000052e-58 < v < 5.19999999999999993e-103

   1. Initial program 99.7%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{{v}^{2} - \frac{98}{5} \cdot H}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\sqrt{\frac{1}{{v}^{2} - \frac{98}{5} \cdot H}} \cdot v\right)} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{{v}^{2} + \left(\mathsf{neg}\left(\frac{98}{5}\right)\right) \cdot H}}} \cdot v\right) \]

      3. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{1}{{v}^{2} + \color{blue}{\frac{-98}{5}} \cdot H}} \cdot v\right) \]

      4. +-commutative N/A

         \[\leadsto \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{\frac{-98}{5} \cdot H + {v}^{2}}}} \cdot v\right) \]

      5. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]

      6. lower-atan.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(v \cdot \sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}\right)} \]

      7. *-commutative N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}} \cdot v\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}} \cdot v\right)} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \tan^{-1} \left(\color{blue}{\sqrt{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}} \cdot v\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{\frac{1}{\frac{-98}{5} \cdot H + {v}^{2}}}} \cdot v\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \tan^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\frac{-98}{5}, H, {v}^{2}\right)}}} \cdot v\right) \]

      12. unpow2 N/A

          \[\leadsto \tan^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{-98}{5}, H, \color{blue}{v \cdot v}\right)}} \cdot v\right) \]

      13. lower-*.f64 99.6

          \[\leadsto \tan^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(-19.6, H, \color{blue}{v \cdot v}\right)}} \cdot v\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(-19.6, H, v \cdot v\right)}} \cdot v\right)} \]

   6. Taylor expanded in v around 0

      \[\leadsto \tan^{-1} \left(\sqrt{\frac{\frac{-5}{98}}{H}} \cdot v\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.2%

      \[\leadsto \tan^{-1} \left(\sqrt{\frac{-0.05102040816326531}{H}} \cdot v\right) \]

   if 5.19999999999999993e-103 < v

   1. Initial program 56.0%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 85.4%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -3.7 \cdot 10^{-141}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -3.7e-141) (atan (fma -9.8 (/ H (* v v)) -1.0)) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -3.7e-141) {
		tmp = atan(fma(-9.8, (H / (v * v)), -1.0));
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -3.7e-141)
		tmp = atan(fma(-9.8, Float64(H / Float64(v * v)), -1.0));
	else
		tmp = atan(1.0);
	end
	return tmp
end

Wolfram

code[v_, H_] := If[LessEqual[v, -3.7e-141], N[ArcTan[N[(-9.8 * N[(H / N[(v * v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -3.7e-141

   1. Initial program 61.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} - \color{blue}{1 \cdot 1}\right) \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)} \]

      3. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1} \cdot 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(\frac{-49}{5} \cdot \frac{H}{{v}^{2}} + \color{blue}{-1}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{{v}^{2}}, -1\right)\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \color{blue}{\frac{H}{{v}^{2}}}, -1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(\frac{-49}{5}, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

      8. lower-*.f64 82.9

         \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(-9.8, \frac{H}{\color{blue}{v \cdot v}}, -1\right)\right) \]

   5. Applied rewrites 82.9%

      \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(-9.8, \frac{H}{v \cdot v}, -1\right)\right)} \]

   if -3.7e-141 < v

   1. Initial program 71.5%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 57.1%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq -2.05 \cdot 10^{-299}:\\ \;\;\;\;\tan^{-1} -1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (v H)
 :precision binary64
 (if (<= v -2.05e-299) (atan -1.0) (atan 1.0)))

C

double code(double v, double H) {
	double tmp;
	if (v <= -2.05e-299) {
		tmp = atan(-1.0);
	} else {
		tmp = atan(1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, h)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    real(8) :: tmp
    if (v <= (-2.05d-299)) then
        tmp = atan((-1.0d0))
    else
        tmp = atan(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double H) {
	double tmp;
	if (v <= -2.05e-299) {
		tmp = Math.atan(-1.0);
	} else {
		tmp = Math.atan(1.0);
	}
	return tmp;
}

Python

def code(v, H):
	tmp = 0
	if v <= -2.05e-299:
		tmp = math.atan(-1.0)
	else:
		tmp = math.atan(1.0)
	return tmp

Julia

function code(v, H)
	tmp = 0.0
	if (v <= -2.05e-299)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, H)
	tmp = 0.0;
	if (v <= -2.05e-299)
		tmp = atan(-1.0);
	else
		tmp = atan(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, H_] := If[LessEqual[v, -2.05e-299], N[ArcTan[-1.0], $MachinePrecision], N[ArcTan[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -2.05e-299

   1. Initial program 67.6%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around -inf

      \[\leadsto \tan^{-1} \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.1%

      \[\leadsto \tan^{-1} \color{blue}{-1} \]

   if -2.05e-299 < v

   1. Initial program 67.1%

      \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \tan^{-1} \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 65.5%

      \[\leadsto \tan^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \tan^{-1} -1 \end{array} \]

FPCore

(FPCore (v H) :precision binary64 (atan -1.0))

C

double code(double v, double H) {
	return atan(-1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(v, h)
use fmin_fmax_functions
    real(8), intent (in) :: v
    real(8), intent (in) :: h
    code = atan((-1.0d0))
end function

Java

public static double code(double v, double H) {
	return Math.atan(-1.0);
}

Python

def code(v, H):
	return math.atan(-1.0)

Julia

function code(v, H)
	return atan(-1.0)
end

MATLAB

function tmp = code(v, H)
	tmp = atan(-1.0);
end

Wolfram

code[v_, H_] := N[ArcTan[-1.0], $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\tan^{-1} \left(\frac{v}{\sqrt{v \cdot v - \left(2 \cdot 9.8\right) \cdot H}}\right) \]
2. Add Preprocessing

3. Taylor expanded in v around -inf

   \[\leadsto \tan^{-1} \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 35.4%

   \[\leadsto \tan^{-1} \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(FPCore (v H)
  :name "Optimal throwing angle"
  :precision binary64
  (atan (/ v (sqrt (- (* v v) (* (* 2.0 9.8) H))))))